Some generalizations of the classical moment problem (Q1865911)

The classical power moment problem (also called Hamburger's moment problem) means to find conditions on sequences of real numbers \(s=(s_n)_{n=0}^\infty\), implying the existence of a positive Borel measure \(\rho\) on the real line such that \(s_n=\int\lambda^n d\rho(\lambda),\;n=0,1,\dots\) There are several, already classical approaches to this problem, including Hamburger's one. In the paper under review, the author's investigation is based on the theory of generalized eigenfunction expansions for selfadjoint operators, namely for those naturally related to the problem. This method leads to two generalizations. Firstly, the sequence \((\lambda_n)\) is replaced by a sequence of polynomials \((P_n(\lambda))\), derived from a certain Jacobi matrix. Secondly, the sequence \((\lambda^n)\) is replaced by a sequence \((\lambda^{\otimes n})\), where \(\lambda\) is a real generalized function. The second generalization is, in fact, an infinite-dimensional moment problem, useful, in particular, for statistical physics.